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A056944
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Amount by which used area of rectangle needed to enclose a non-touching spiral of length n on a square lattice exceeds unused area.
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7
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0, 1, 2, 2, 2, 4, 3, 2, 4, 6, 4, 2, 4, 6, 8, 5, 2, 4, 6, 8, 10, 6, 2, 4, 6, 8, 10, 12, 7, 2, 4, 6, 8, 10, 12, 14, 8, 2, 4, 6, 8, 10, 12, 14, 16, 9, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 11, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 12, 2, 4, 6, 8, 10, 12, 14, 16
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OFFSET
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0,3
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COMMENTS
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m (when n is m-th triangular number) followed by m even numbers from 2 through 2m.
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LINKS
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FORMULA
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a(n) = 2n - floor((sqrt(8n+1)-1)/2)*ceiling((sqrt(8n+1)-1)/2) = 2n - A002024(n)*A003056(n) = 2n - A056942(n) = n -A056943(n). If n = t(t+1)/2 then a(n)=t; if n = t(t+1)/2+k with 0 < k <= t then a(n)=2k.
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EXAMPLE
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a(9)=6 since spiral is as marked by 9 X's in 4*3 = 12 rectangle, with 12-9 = 3 spaces unused and a used-unused difference of 9-3 = 6:
X.XX
X..X
XXXX
As a triangle, the first few rows are: 1; 2, 2; 2, 4, 3; 2, 4, 6, 4; 2, 4, 6, 8, 5; 2, 4, 6, 8, 10, 6; 2, 4, 6, 8, 10, 12, 7; ... (= reversal of triangle A143595). Row sums = n^2. - Gary W. Adamson, Aug 26 2008
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MATHEMATICA
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uar[n_]:=Module[{c=(Sqrt[8n+1]-1)/2}, 2n-Floor[c]Ceiling[c]]; Array[uar, 90, 0] (* Harvey P. Dale, Aug 14 2013 *)
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PROG
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(Magma) [2*n-Floor((Sqrt(8*n+1)-1)/2)*Ceiling((Sqrt(8*n+1)-1)/2): n in [0..90]]; // Vincenzo Librandi, Aug 06 2017
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CROSSREFS
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KEYWORD
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easy,nonn,nice
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AUTHOR
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STATUS
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approved
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